Prerequisites for Calculus

CHAPTER Prerequisites for Calculus. Lines. Functions and Graphs.3 Eponential Functions.4 Parametric Equations.5 Functions and Logarithms Eponential functions are used to model situations in which growth or decay changes dramatically. Such situations are found in nuclear power plants, which contain rods of plutonium-39, an etremely toic radioactive isotope. Operating at full capacity for one year, a 000-megawatt power plant discharges about 435 lb of plutonium-39. With a half-life of 4,400 years, how much of the isotope will remain after 000 years? This question can be answered with the mathematics covered in Section.3..6 Trigonometric Functions

Section. Lines 3 CHAPTER Overview This chapter reviews the most important things you need to know to start learning calculus. It also introduces the use of a graphing utility as a tool to investigate mathematical ideas, to support analytic work, and to solve problems with numerical and graphical methods. The emphasis is on functions and graphs, the main building blocks of calculus. Functions and parametric equations are the major tools for describing the real world in mathematical terms, from temperature variations to planetary motions, from brain waves to business cycles, and from heartbeat patterns to population growth. Many functions have particular importance because of the behavior they describe. Trigonometric functions describe cyclic, repetitive activity; eponential, logarithmic, and logistic functions describe growth and decay; and polynomial functions can approimate these and most other functions.. Lines What you will learn about... Increments Slope of a Line Parallel and Perpendicular Lines Equations of Lines Applications and why... Linear equations are used etensively in business and economic applications. Increments One reason calculus has proved to be so useful is that it is the right mathematics for relating the rate of change of a quantity to the graph of the quantity. Eplaining that relationship is one goal of this book. It all begins with the slopes of lines. When a particle in the plane moves from one point to another, the net changes or increments in its coordinates are found by subtracting the coordinates of its starting point from the coordinates of its stopping point. DEFINITION Increments If a particle moves from the point (, y ) to the point (, y ), the increments in its coordinates are = - and y = y - y. The symbols and y are read delta and delta y. The letter is a Greek capital d for difference. Neither nor y denotes multiplication; is not delta times nor is y delta times y. Increments can be positive, negative, or zero, as shown in Eample. y L P (, y ) y rise P (, y ) Q(, y ) run EXAMPLE Finding Increments The coordinate increments from (4, -3) to (, 5) are = - 4 = -, y = 5 - (-3) = 8. From (5, 6) to (5, ), the increments are = 5-5 = 0, y = - 6 = -5. Now Try Eercise. O Figure. The slope of line L is m = rise run = y. Slope of a Line Each nonvertical line has a slope, which we can calculate from increments in coordinates. Let L be a nonvertical line in the plane and P (, y ) and P (, y ) two points on L (Figure.). We call y = y - y the rise from P to P and = - the run from

4 Chapter Prerequisites for Calculus P to P. Since L is not vertical, Z 0 and we define the slope of L to be the amount of rise per unit of run. It is conventional to denote the slope by the letter m. L L DEFINITION Slope Slope m Slope m m θ θ m Let P (, y ) and P (, y ) be points on a nonvertical line, L. The slope of L is m = rise. run = y = y - y - Figure. If L 7 L, then u = u and m = m. Conversely, if m = m, then u = u and L 7 L. O y L C L Slope m Slope m h A D a B Figure.3 ^ ADC is similar to ^ CDB. Hence f is also the upper angle in ^ CDB, where tan f = a>h. 6 5 4 3 y Along this line,. Along this line, y 3. (, 3) 0 3 4 Figure.4 The standard equations for the vertical and horizontal lines through the point (, 3) are = and y = 3. (Eample ) A line that goes uphill as increases has a positive slope. A line that goes downhill as increases has a negative slope. A horizontal line has slope zero since all of its points have the same y-coordinate, making y = 0. For vertical lines, = 0 and the ratio y> is undefined. We epress this by saying that vertical lines have no slope. Parallel and Perpendicular Lines Parallel lines form equal angles with the -ais (Figure.). Hence, nonvertical parallel lines have the same slope. Conversely, lines with equal slopes form equal angles with the -ais and are therefore parallel. If two nonvertical lines L and L are perpendicular, their slopes m and m satisfy m m = -, so each slope is the negative reciprocal of the other: The argument goes like this: In the notation of Figure.3, m = tan f = a>h, while m = tan f = -h>a. Hence, m m = (a>h)(-h>a) = -. Equations of Lines The vertical line through the point ( a, b) has equation = a since every -coordinate on the line has the value a. Similarly, the horizontal line through ( a, b) has equation y = b. EXAMPLE Finding Equations of Vertical and Horizontal Lines The vertical and horizontal lines through the point (, 3) have equations = and y = 3, respectively (Figure.4). Now Try Eercise 9. We can write an equation for any nonvertical line L if we know its slope m and the coordinates of one point P (, y ) on it. If P(, y) is any other point on L, then so that DEFINITION The equation m = - y - y = m( - ) Point-Slope Equation, m m = -. m y - y - = m, y = m( - ) + y or y = m( - ) + y. is the point-slope equation of the line through the point (, y ) with slope m.

Section. Lines 5 EXAMPLE 3 Using the Point-Slope Equation Write the point-slope equation for the line through the point (, 3) with slope -3>. We substitute =, y = 3, and m = -3> into the point-slope equation and obtain y = - 3 ( - ) + 3 or y = - 3 + 6. Now Try Eercise 3. (0, b) y (, y) y = m + b Slope m The y-coordinate of the point where a nonvertical line intersects the y-ais is the y-intercept of the line. Similarly, the -coordinate of the point where a nonhorizontal line intersects the -ais is the -intercept of the line. A line with slope m and y-intercept b passes through (0, b) (Figure.5), so y = m( - 0) + b, or, more simply, y = m + b. 0 b Figure.5 A line with slope m and y-intercept b. DEFINITION Slope-Intercept Equation The equation y = m + b is the slope-intercept equation of the line with slope m and y-intercept b. EXAMPLE 4 Writing the Slope-Intercept Equation Write the slope-intercept equation for the line through (-, -) and (3, 4). The line s slope is m = 4 - (-) 3 - (-) = 5 5 =. We can use this slope with either of the two given points in the point-slope equation. For (, y ) = (-, -), we obtain y = # ( - (-)) + (-) y = + + (-) y = +. Now Try Eercise 7. If A and B are not both zero, the graph of the equation A + By = C is a line. Every line has an equation in this form, even lines with undefined slopes. DEFINITION General Linear Equation The equation A + By = C (A and B not both 0) is a general linear equation in and y.

6 Chapter Prerequisites for Calculus Although the general linear form helps in the quick identification of lines, the slopeintercept form is the one to enter into a calculator for graphing. y = 8 + 4 5 Figure.6 The line (Eample 5) [ 5, 7] by [, 6] 8 + 5y = 0. EXAMPLE 5 Analyzing and Graphing a General Linear Equation Find the slope and y-intercept of the line 8 + 5y = 0. Graph the line. Solve the equation for y to put the equation in slope-intercept form: 8 + 5y = 0 5y = -8 + 0 y = - 8 5 + 4 This form reveals the slope (m = -8>5) and y-intercept (b = 4), and puts the equation in a form suitable for graphing (Figure.6). Now Try Eercise 7. EXAMPLE 6 Writing Equations for Lines Write an equation for the line through the point (-, ) that is (a) parallel, and (b) perpendicular to the line L: y = 3-4. The line L, y = 3-4, has slope 3. (a) The line y = 3( + ) +, or y = 3 + 5, passes through the point (-, ), and is parallel to L because it has slope 3. (b) The line y = (->3)( + ) +, or y = (->3) + 5>3, passes through the point (-, ), and is perpendicular to L because it has slope ->3. Now Try Eercise 3. EXAMPLE 7 Determining a Function The following table gives values for the linear function f() = m + b. Determine m and b. - f() 4>3-4>3-3>3 The graph of f is a line. From the table we know that the following points are on the line: (-, 4>3), (, -4>3), (, -3>3). Using the first two points, the slope m is m = -4>3 - (4>3) - (-) = -6 = -3. So f() = -3 + b. Because f(-) = 4>3, we have f(-) = -3(-) + b 4>3 = 3 + b b = 5>3. continued

Section. Lines 7 Thus, m = -3, b = 5>3, and f() = -3 + 5>3. We can use either of the other two points determined by the table to check our work. Now Try Eercise 35. Applications Many important variables are related by linear equations. For eample, the relationship between Fahrenheit temperature and Celsius temperature is linear, a fact we use to advantage in the net eample. EXAMPLE 8 Temperature Conversion Find the relationship between Fahrenheit and Celsius temperature. Then find the Celsius equivalent of 90 F and the Fahrenheit equivalent of -5 C. Because the relationship between the two temperature scales is linear, it has the form F = mc + b. The freezing point of water is F = 3 or C = 0, while the boiling point is F = or C = 00. Thus, 3 = m # 0 + b and = m # 00 + b, so b = 3 and m = ( - 3)>00 = 9>5. Therefore, F = 9 5 C + 3, or C = 5 (F - 3). 9 These relationships let us find equivalent temperatures. The Celsius equivalent of 90 F is The Fahrenheit equivalent of -5 C is C = 5 (90-3) L 3.. 9 F = 9 (-5) + 3 = 3. 5 Now Try Eercise 43. Some graphing utilities have a feature that enables them to approimate the relationship between variables with a linear equation. We use this feature in Eample 9. TABLE. World Population Year Population (millions) 980 4454 985 4853 990 585 995 5696 003 6305 004 6378 005 6450 Source: U.S. Bureau of the Census, Statistical Abstract of the United States, 004 005 and 00. It can be difficult to see patterns or trends in lists of paired numbers. For this reason, we sometimes begin by plotting the pairs (such a plot is called a scatter plot) to see whether the corresponding points lie close to a curve of some kind. If they do, and if we can find an equation y = f() for the curve, then we have a formula that. summarizes the data with a simple epression, and. lets us predict values of y for other values of. The process of finding a curve to fit data is called regression analysis and the curve is called a regression curve. There are many useful types of regression curves power, eponential, logarithmic, sinusoidal, and so on. In the net eample, we use the calculator s linear regression feature to fit the data in Table. with a line. EXAMPLE 9 Regression Analysis Predicting World Population Starting with the data in Table., build a linear model for the growth of the world population. Use the model to predict the world population in the year 05, and compare this prediction with the Statistical Abstract prediction of 79 million. continued

8 Chapter Prerequisites for Calculus Why Not Round the Decimals in Equation Even More? If we do, our final calculation will be way off. Using y = 80-53,849, for instance, gives y = 735 when = 05, as compared to y = 765, an increase of 86 million. The rule is: Retain all decimal places while working a problem. Round only at the end. We rounded the coefficients in Equation enough to make it readable, but not enough to hurt the outcome. However, we knew how much we could safely round only from first having done the entire calculation with numbers unrounded. Model Upon entering the data into the grapher, we find the regression equation to be approimately y = 79.957-53848.76, where represents the year and y the population in millions. Figure.7a shows the scatter plot for Table. together with a graph of the regression line just found. You can see how well the line fits the data. () X = 05 Y = 764.98 [975, 00] by [4000, 7500] (a) Figure.7 (Eample 9) [975, 00] by [4000, 7500] (b) Solve Graphically Our goal is to predict the population in the year 05. Reading from the graph in Figure.7b, we conclude that when is 05, y is approimately 765. Rounding Rule Round your answer as appropriate, but do not round the numbers in the calculations that lead to it. Confirm Algebraically Evaluating Equation for = 05 gives y = 79.957(05) - 53848.76 L 765. Interpret The linear regression equation suggests that the world population in the year 05 will be about 765 million, or approimately 36 million more than the Statistical Abstract prediction of 79 million. Now Try Eercise 45. Regression Analysis Regression analysis has four steps:. Plot the data (scatter plot).. Find the regression equation. For a line, it has the form y = m + b. 3. Superimpose the graph of the regression equation on the scatter plot to see the fit. 4. Use the regression equation to predict y-values for particular values of.

Section. Lines 9 Quick Review. (For help, go to Section..) Eercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator.. Find the value of y that corresponds to = 3 in y = - + 4( - 3).. Find the value of that corresponds to y = 3 in y = 3 - ( + ). In Eercises 3 and 4, find the value of m that corresponds to the values of and y. 3. = 5, y =, m = y - 3-4 4. = -, y = -3, m = - y 3 - In Eercises 5 and 6, determine whether the ordered pair is a solution to the equation. 5. 3-4y = 5 6. y = - + 5 (a) (, > 4 ) (b) (3, -) (a) (-, 7) (b) (-, ) In Eercises 7 and 8, find the distance between the points. 7. (, 0 ), ( 0, ) 8. (, ), (, ->3) In Eercises 9 and 0, solve for y in terms of. 9. 4-3y = 7 0. - + 5y = -3 Section. Eercises Eercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Eercises 4, find the coordinate increments from A to B.. A(, ), B(-, -). A(-3, ), B(-, -) 3. A(-3, ), B(-8, ) 4. A( 0, 4 ), B( 0, ) In Eercises 5 8, let L be the line determined by points A and B. (a) Plot A and B. (b) Find the slope of L. (c) Draw the graph of L. 5. A(, -), B(, ) 6. A(-, -), B(, -) 7. A(, 3 ), B(-, 3) 8. A(, ), B(, -3) In Eercise 9, write an equation for (a) the vertical line and (b) the horizontal line through the point P. 9. P( 3, ) 0. P(-, 4>3). P(0, - ). P(-p, 0) In Eercises 3 6, write the point-slope equation for the line through the point P with slope m. 3. P(, ), m = 4. P(-, ), m = - 5. P( 0, 3 ), m = 6. P(-4, 0), m = - In Eercises 7 0, write the slope-intercept equation for the line with slope m and y-intercept b. 7. m = 3, b = - 8. m = -, b = 9. m = ->, b = -3 0. m = >3, b = - In Eercises 4, write a general linear equation for the line through the two points.. ( 0, 0 ), (, 3). (, ), (, ) 3. (-, 0), (-, -) 4. (-, ), (, -) In Eercises 5 and 6, the line contains the origin and the point in the upper right corner of the grapher screen. Write an equation for the line. 5. 6. [ 0, 0] by [ 5, 5] [ 5, 5] by [, ] In Eercises 7 30, find the (a) slope and (b) y-intercept, and (c) graph the line. 7. 3 + 4y = 8. + y = 9. 30. y = + 4 3 + y 4 = In Eercises 3 34, write an equation for the line through P that is (a) parallel to L, and (b) perpendicular to L. 3. P(0, 0), L: y = - + 3. P(-, ), 33. P(-, 4), L: = 5 34. P(-, >), In Eercises 35 and 36, a table of values is given for the linear function f() = m + b. Determine m and b. 35. 36. f() 3 9 5 6 L: + y = 4 L: y = 3 f() - 4-4 6-7

0 Chapter Prerequisites for Calculus In Eercises 37 and 38, find the value of or y for which the line through A and B has the given slope m. 37. A(-, 3), B( 4, y ), m = ->3 38. A(-8, -), B, ( ), m = 39. Revisiting Eample 4 Show that you get the same equation in Eample 4 if you use the point ( 3, 4) to write the equation. 40. Writing to Learn - and y-intercepts (a) Eplain why c and d are the -intercept and y-intercept, respectively, of the line c + y d =. (b) How are the -intercept and y-intercept related to c and d in the line c + y d =? 4. Parallel and Perpendicular Lines For what value of k are the two lines + ky = 3 and + y = (a) parallel? (b) perpendicular? Group Activity In Eercises 4 44, work in groups of two or three to solve the problem. 4. Insulation By measuring slopes in the figure below, find the temperature change in degrees per inch for the following materials. (a) gypsum wallboard (b) fiberglass insulation (c) wood sheathing (d) Writing to Learn Which of the materials in (a) (c) is the best insulator? the poorest? Eplain. Temperature ( F) 80 70 60 50 40 30 0 0 0 Air inside room at 7 F 0 Gypsum wallboard Fiberglass between studs 3 4 5 6 7 Distance through wall (inches) Sheathing Siding Air outside at 0 F 43. Pressure Under Water The pressure p eperienced by a diver under water is related to the diver s depth d by an equation of the form p = kd + (k a constant). When d = 0 meters, the pressure is atmosphere. The pressure at 00 meters is 0.94 atmospheres. Find the pressure at 50 meters. 44. Modeling Distance Traveled A car starts from point P at time t = 0 and travels at 45 mph. (a) Write an epression d( t) for the distance the car travels from P. (b) Graph y = d(t). (c) What is the slope of the graph in (b)? What does it have to do with the car? (d) Writing to Learn Create a scenario in which t could have negative values. (e) Writing to Learn Create a scenario in which the y-intercept of y = d(t) could be 30. In Eercises 45 and 46, use linear regression analysis. 45. Table. shows the mean annual compensation of construction workers. TABLE. Construction Workers Average Annual Compensation Year Annual Total Compensation (dollars) 999 4,598 000 44,764 00 47,8 00 48,966 Source: U.S. Bureau of the Census, Statistical Abstract of the United States, 004 005. (a) Find the linear regression equation for the data. (b) Find the slope of the regression line. What does the slope represent? (c) Superimpose the graph of the linear regression equation on a scatter plot of the data. (d) Use the regression equation to predict the construction workers average annual compensation in the year 008. 46. Table.3 lists the ages and weights of nine girls. TABLE.3 Girls Ages and Weights Age (months) Weight (pounds) 9 3 4 5 7 8 9 3 3 8 34 3 38 34 43 39 (a) Find the linear regression equation for the data. (b) Find the slope of the regression line. What does the slope represent? (c) Superimpose the graph of the linear regression equation on a scatter plot of the data. (d) Use the regression equation to predict the approimate weight of a 30-month-old girl.

Section. Lines Standardized Test Questions 47. True or False The slope of a vertical line is zero. Justify your answer. 48. True or False The slope of a line perpendicular to the line y = m + b is > m. Justify your answer. 49. Multiple Choice Which of the following is an equation of the line through (-3, 4) with slope >? (A) y - 4 = ( + 3) (C) y - 4 = -( + 3) (E) y + 3 = ( - 4) (B) y + 3 = ( - 4) (D) y - 4 = ( + 3) 50. Multiple Choice Which of the following is an equation of the vertical line through (, 4 )? (A) y = 4 (B) = (C) y = -4 (D) = 0 (E) = - 5. Multiple Choice Which of the following is the -intercept of the line y = - 5? (A) = -5 (B) = 5 (C) = 0 (D) = 5> (E) = -5> 5. Multiple Choice Which of the following is an equation of the line through (-, -) parallel to the line y = -3 +? 54. Fahrenheit Versus Celsius We found a relationship between Fahrenheit temperature and Celsius temperature in Eample 8. (a) Is there a temperature at which a Fahrenheit thermometer and a Celsius thermometer give the same reading? If so, what is it? (b) Writing to Learn Graph y = (9>5) + 3, y = (5>9)( - 3), and y 3 = in the same viewing window. Eplain how this figure is related to the question in part (a). 55. Parallelogram Three different parallelograms have vertices at (-, ), (, 0), and (, 3). Draw the three and give the coordinates of the missing vertices. 56. Parallelogram Show that if the midpoints of consecutive sides of any quadrilateral are connected, the result is a parallelogram. 57. Tangent Line Consider the circle of radius 5 centered at (0, 0). Find an equation of the line tangent to the circle at the point (3, 4). 58. Group Activity Distance from a Point to a Line This activity investigates how to find the distance from a point P( a, b) to a line L: A + By = C. (a) Write an equation for the line M through P perpendicular to L. (b) Find the coordinates of the point Q in which M and L intersect. (c) Find the distance from P to Q. (A) y = -3 + 5 (B) y = -3-7 (C) (D) y = -3 + (E) y = -3-4 y = 3-3 Etending the Ideas 53. The median price of eisting single-family homes has increased consistently during the past few years. However, the data in Table.4 show that there have been differences in various parts of the country. TABLE.4 Median Price of Single-Family Homes Year South (dollars) West (dollars) 999 45,900 73,700 000 48,000 96,400 00 55,400 3,600 00 63,400 38,500 003 68,00 60,900 Source: U.S. Bureau of the Census, Statistical Abstract of the United States, 004 005. (a) Find the linear regression equation for home cost in the South. (b) What does the slope of the regression line represent? (c) Find the linear regression equation for home cost in the West. (d) Where is the median price increasing more rapidly, in the South or the West?